Madeleine Peyroux's street-singing days are far behind as she swings into stardom in the jazz world

O.K., sing a song for me right now." The scene was a sidewalk in Paris in 1990. The challenge was issued by the leader of a ragtag touring group called the Lost Wandering Blues & Jazz Band, who was holding an impromptu audition. The wannabe was an American-born teenager named Madeleine Peyroux, who had been busking around Paris. She crooned Jeepers Creepers, snapping her fingers as accompaniment, and got the job.

It's a long way from street singing to stardom, and the intervening 15 years have been riddled with delays and detours, but Peyroux, 31, has made it. With two successful CDs...